P(x)=x(10-0.5x)

Solution for P(x)=x(10-0.5x) equation:

(P)=P(10-0.5P)

We move all terms to the left:

(P)-(P(10-0.5P))=0

We add all the numbers together, and all the variables

P-(P(-0.5P+10))=0


We calculate terms in parentheses: -(P(-0.5P+10)), so:

P(-0.5P+10)

We multiply parentheses

0P^2+10P

We add all the numbers together, and all the variables

P^2+10P

Back to the equation:

-(P^2+10P)


We get rid of parentheses

-P^2+P-10P=0

We add all the numbers together, and all the variables

-1P^2-9P=0

a = -1; b = -9; c = 0;
Δ = b2-4ac
Δ = -92-4·(-1)·0
Δ = 81
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{81}=9$
$P_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-9)-9}{2*-1}=\frac{0}{-2}
=0
$
$P_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-9)+9}{2*-1}=\frac{18}{-2}
=-9
$